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Pre-Calculus Review
What is a function?

Graphically

Algebraically

Numerically

Verbally
Examples of relations that are and are not functions
Evaluate a function using your example: f(6) =
f(-2) =
Independent vs. Dependent Variables:
One-to-one functions:

Graphically

Algebraically

Numerically

Verbally
Types of Functions
1. Polynomial functions (subset: power functions which have only one term)

Linear : Basic power function:
Slope (def., formula)
Domain:
Range:
slope of Parallel lines are
slope of Perpendicular (Normal) lines are
Forms of linear equations:
Slope intercept form
Standard form
Point-Slope form
2

Quadratic: Basic power function:
Similarly shaped functions:
Domain:
Range:
Even or Odd?
Why? Algebraically:
Graphically:
****Graphs have symmetry. Functions are even, odd or neither. ****
On what interval is the quadratic function increasing?
Decreasing?
**** Interval notation: ( means not included, [means included, never include infinity****

Cubic: Basic power function:
Similarly shaped functions:
Domain:
Range:
Even or Odd?
Why? Algebraically:
Graphically:
On what interval is the function increasing?
Determining Symmetry





Polynomial functions with all even degrees are even
Polynomial functions with all odd degrees are odd
Polynomial functions with mixed odd/even degrees are neither
Constants are of even degrees
Any function that is more complicated, do f(-x) test
Examples: do the f(-x) test on the following:
𝑓(𝑥) = 𝑥 4 + 𝑥 2
𝑓(𝑥) = 𝑥 3 + 𝑥 2
Decreasing?
3

Radical : Basic power function:
Domain:
Range:
***Can only take even roots of non-negative numbers (zero or positive). ***
Similarly shaped Functions
𝑓(𝑥) = 𝑥 𝑎/𝑏
If b is even, graph Does Not Exist when x <0.
If b is odd, and a is even, graph has even symmetry.
If b is odd, and a is odd, graph has odd symmetry.
Generalizing: 𝒇(𝒙) = 𝒌𝒙𝒂
I
II
a<1
III
0<a<1
a=1
a>1
IV
𝑔(𝑥)
2. Rational Functions: ratio of 2 polynomial functions 𝑓(𝑥) = ℎ(𝑥)
***Denominators cannot equal zero***
Basic function: Reciprocal function:
1
𝑦=𝑥
Domain:
Range:
Horizontal asymptote:
Vertical asymptote:
***Asymptotes are lines represented by equations, not a lone number. ***
4
o
o
Vertical asymptotes describe where the y - values are approaching positive or negative infinity
from the left or the right.
Factors that cancel out indicate holes; factors that remain in the denominator indicate vertical
asymptotes.
Example: 𝑓(𝑥)
=
𝑥−1
𝑥 2 −4𝑥+3
=
Hole at
Vertical asymptote:
Domain:
o
o
o
Horizontal asymptotes describe end behavior, what the y-values are approaching as x approaches
positive or negative infinity.
Graphs can cross a horizontal asymptote.
For rational functions, compare the degrees of the numerator and denominator.
“BOSTON”
Bottom degree is larger
O (zero) is the location of the horizontal asymptote
Same degrees
Take the coefficients as the asymptote
Otherwise
No horizontal asymptote
Example continued from above: 𝑓(𝑥)
=
𝑥−1
𝑥 2 −4𝑥+3
example:
𝑓(𝑥) =
5𝑥 4 −3
3𝑥 4 +2𝑥
Horizontal asymptote:
Horizontal asymptote:
Range:
Vertical asymptote:
5
3. Absolute value Function
Basic Equation:
Domain:
Range:
Even or Odd?
|4| =
|−4| =
|𝑎| = {
Absolute value functions can be written as a piecewise function:
𝑦 = |𝑥| = {
𝑦 = |2𝑥 − 5| = {
Examples:
Write a piecewise function for the graph.
Y=
1 − 𝑥, 𝑖𝑓 𝑥 ≤ 1
Graph 𝑓(𝑥) = { 2
𝑥 , 𝑖𝑓 𝑥 > 1
Domain:
Domain:
Range:
Range: